Question: Simplify and expand the following expression: $ \dfrac{2}{x + 1}+ \dfrac{4}{5x - 10}- \dfrac{2x}{x^2 - x - 2} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the second term: $ \dfrac{4}{5x - 10} = \dfrac{4}{5(x - 2)}$ We can factor the quadratic in the third term: $ \dfrac{2x}{x^2 - x - 2} = \dfrac{2x}{(x + 1)(x - 2)}$ Now we have: $ \dfrac{2}{x + 1}+ \dfrac{4}{5(x - 2)}- \dfrac{2x}{(x + 1)(x - 2)} $ The least common multiple of the denominators is: $ (x + 1)(x - 2)$ In order to get the first term over $(x + 1)(x - 2)$ , multiply by $\dfrac{5(x - 2)}{5(x - 2)}$ $ \dfrac{2}{x + 1} \times \dfrac{5(x - 2)}{5(x - 2)} = \dfrac{10(x - 2)}{(x + 1)(x - 2)} $ In order to get the second term over $(x + 1)(x - 2)$ , multiply by $\dfrac{x + 1}{x + 1}$ $ \dfrac{4}{5(x - 2)} \times \dfrac{x + 1}{x + 1} = \dfrac{4(x + 1)}{(x + 1)(x - 2)} $ In order to get the third term over $(x + 1)(x - 2)$ , multiply by $\dfrac{5}{5}$ $ \dfrac{2x}{(x + 1)(x - 2)} \times \dfrac{5}{5} = \dfrac{10x}{(x + 1)(x - 2)} $ Now we have: $ \dfrac{10(x - 2)}{(x + 1)(x - 2)} + \dfrac{4(x + 1)}{(x + 1)(x - 2)} - \dfrac{10x}{(x + 1)(x - 2)} $ $ = \dfrac{ 10(x - 2) + 4(x + 1) - 10x} {(x + 1)(x - 2)} $ Expand: $ = \dfrac{10x - 20 + 4x + 4 - 10x}{5x^2 - 5x - 10} $ $ = \dfrac{4x - 16}{5x^2 - 5x - 10}$